1. Introduction
Over the past two decades, tremendous attention has been paid to the cooperative control of swarm systems, which has led to fruitful research results [
1,
2,
3], and has found broad applications in the area of unmanned swarm systems, such as collaborative searching, escorting, localization and mapping by unmanned aerial vehicles [
4,
5,
6] and unmanned mobile robots [
7,
8,
9]. In particular, the motion in formation for unmanned swarm systems is the foundation of many tasks, where all of the individuals in the swarm are supposed to cooperate to maintain some desired relative position and/or orientation with respect to each other while following some common reference path. The existing control approaches dealing with the formation problem can be classified roughly into three categories [
10]. The first category is the behavior-based approach, which was first proposed by Balch and Arkin in [
11] for four kinds of formation, namely, line, column, wedge and diamond. The guidance and formation are fulfilled by designing and weighting motor schemas. Typical works on the behavior-based approach can be further found in [
12,
13]. The second category is the leader–follower approach, which defines the global formation by a series of pre-specified leader-following tracking problems. Since the classic leader-following tracking problem has been well treated in the literature, the leader–follower approach has become an effective way to solve the formation problem [
14,
15]. The third category is the virtual structure approach, where the formation is defined by the static or dynamic relative position/orientation of each agent with respect to a virtual leader. Recently, owing to the application of graph theory in the control of multi-agent systems, the virtual structure approach has become the most popular control strategy for distributed formation control over sparse and vulnerable communication networks [
16].
The rapid development of autonomous underwater vehicles (AUVs) has greatly enabled underwater tasks, such as environment inspection, military surveillance, oceanographic observation, site searching and so on [
17,
18,
19]. In contrast to a single AUV, AUV swarms show a better system reliability against failure and system adaptability for complex tasks [
20,
21,
22]. There have thus far been extensive works devoted to the formation control of an AUV swarm. A group of torpedo-type AUVs was investigated in [
23] under the centralized management by an unmanned surface vessel, constituting a star-like command and communication system, with the unmanned surface vessel being the center. The formation is achieved by assigning each AUV its corresponding reference trajectory. Both references [
24,
25] adopted the leader–follower control structure. Environmental disturbances and input saturation were considered in [
24], tackled by dynamic surface control and a hyperbolic tangent function, respectively. To simultaneously deal with communication delay, packet discreteness and dropout, a curve fitting method was developed by [
25] to make a prediction for the states of the AUV. On the other hand, the virtual structure approach based on graph theory was invoked in [
26,
27,
28,
29,
30,
31]. Li et al. considered prescribed performance control for underactuated AUVs subject to uncertain dynamics and disturbances [
26]. By using radial basis functions, the lumped uncertainties of the entire system were transformed into a linearly parameterized form with a single uncertain parameter. Different from [
26], model predictive control and an extended state observer were employed in [
27] to cope with unknown ocean current disturbances. Time delay issues were discussed in [
28,
29]. Both bounded and unbounded communication delays were considered in [
28], and two separate communication networks for both position and velocity were implemented to realize formation trajectory tracking. A consistent control algorithm was proposed by [
29] to deal with communication delays by Gershgorin disk theorem and Nyquist law. To recover the information of the global trajectory, distributed observers were resorted to in [
30,
31]. A neural-network-based deterministic learning approach was established in [
30] to tackle the nonlinear uncertain dynamics of the AUVs. To reduce the risk of actuator saturation, a hyperbolic tangent function was adopted in [
31] to design a decentralized formation tracking control law that can also handle system nonlinearities and measurement noises.
From the perspective of the system structure, swarm systems can be categorized as a leaderless swarm and leader–follower swarm, where the formation control of swarms is mainly involved with the latter one by viewing the formation command as the leader. The cooperative output regulation theory proves to be an effective control framework tackling the distributed control of leader–follower swarm systems [
32], which can also be viewed as an extension of the virtual structure approach. In the cooperative output regulation framework, the exosystem is viewed as the leader, and each agent is viewed as a follower. The control objectives are twofold: (1) the stability of the closed-loop system should be guaranteed; (2) the output of each agent should track a class of reference signals while rejecting a class of external disturbances. So far, there have been some attempts applying the cooperative output regulation theory to solve the distributed formation problem [
33,
34,
35,
36,
37]. Wang considered a linear swarm system over a static communication network [
33]. By taking the system matrix of the exosystem as prior knowledge, a distributed internal model was constructed for each agent utilizing virtual errors. Hua et al. studied the case where the virtual leader has an external input [
34]. A sign-function-based distributed observer was proposed to recover the state of the virtual leader. Like [
33], the system matrix of the virtual leader should be known in advance so as to obtain the solution to the regulator equations. The situation of multiple virtual leaders was considered in [
35], and the formation problem was integrated with the objective of containment control. Through a thorough analysis on the Laplacian of the associated communication graph, an adaptive distributed control law was proposed to solve the formation problem. Similar to [
33], Li et al. also made use of the virtual-error-based distributed internal model [
36], while both the swarm system and the virtual leader considered in [
36] are nonlinear. Local stability results were obtained by performing Jacobian linearization around the origin of the closed-loop system. In practice, the communication network for the swarm might not always be safe or reliable. Huang and Dong investigated the scenario of false data injection into the communication network [
37]. By adopting linear matrix inequality techniques, a robust output regulation control law was proposed so that the tracking errors can be guaranteed to be uniformly ultimately bounded under uniform feedback quantization.
In this paper, we consider the distributed path tracking problem for a swarm of AUVs modeled by second-order multi-agent systems. The AUV swarm should track a class of polynomial path signals while keeping a desired relative position with respect to each other. Different from the existing results, the application scenario of this paper has three distinguished characteristics. First, the unreliable communication network for the multi-agent system was modeled as a switching graph satisfying the jointly connected condition, which allows the communication network topology to be disconnected the entire time. Second, in an underwater environment, absolute position measurement might not be feasible. To address this issue, in this paper, we consider the case where only the relative position between AUVs can be obtained for trajectory tracking control. Third, in the presence of uncertain mass, inertia and velocity damping, the AUV dynamics are assumed to contain uncertain system parameters. By applying the cooperative output regulation control framework, a novel distributed robust control scheme is proposed to solve the distributed path tracking problem, which consists of three parts. First, to cope with communication network uncertainty, the distributed observer was invoked to recover the polynomial path for each AUV, which decouples the dynamics of the AUVs and the virtual leader, thus making it easy to conceive a feasible distributed control scheme under the unreliable communication environment. Second, based on the relative position measurement between AUVs, a pseudo position estimator was adopted to generate the pseudo position for each AUV. The interesting fact regarding the pseudo position lies in that, though it will not converge to the absolute position of the AUV, the differences between the absolute positions and pseudo positions of all of the AUVs will converge to a common constant vector, which paves the way to the success of the proposed distributed control scheme. Finally, based on the estimated polynomial path and the pseudo position, a certainty equivalent robust internal model control law was synthesized to achieve asymptotic reference trajectory tracking, where the internal model compensator aims to tackle uncertain system parameters. It was rigorously proven that the distributed path tracking problem can be solved by the proposed distributed robust control scheme. In contrast to the existing results, the main contributions of this paper are twofold:
In terms of communication topology, reference [
23] adopted a centralized structure with the unmanned surface vessel as the communication center, whereas [
24,
25] adopted the leader–follower approach, whose communication topology is essentially a pure tree-like structure. In [
26,
27,
28,
29,
30,
31], based on graph theory, the communication topology is relaxed to be freely designed as long as the associated communication graph is connected. However, in all of these works [
26,
27,
28,
29,
30,
31], the communication topology should be connected the entire time, which might be impractical for certain application scenarios. In contrast, in this paper, we allow the communication network to be jointly connected, which can be disconnected for the entire time, thus greatly reducing the requirement imposed on the communication network.
In [
23,
24,
25,
26,
27,
28,
29,
30,
31], the absolute position feedback of the AUV is necessary to stabilize the closed-loop system dynamics so that the control objective of tracking or formation can be fulfilled, while, as indicated by [
18,
20,
22], the global localization of AUVs might be costly in an underwater environment, and might sometimes even be impossible, such as in a deep ocean environment. To address this issue, in this paper, the proposed control method only needs a relative position measurement of the neighboring AUVs over the communication network to achieve reference path tracking and maintain a relative formation, which makes it more practical and cost competitive in contrast to the existing results.
2. Graph Notation
A graph is defined by a node set and an edge set . For , , means that there exists an edge in from node i to node j. If , then node i is called a neighbor of node j. Let denote the neighbor set of node i. If if and only if , then the edge is called undirected. If all of the edges of a graph are undirected, then the graph is called undirected. If contains a set of edges of the form , then the set is called a path of from node to node , and node is said to be reachable from node . A graph is said to contain a spanning tree if there exists a node in such that all of the other nodes are reachable from it, and this node is called the root of the spanning tree. Given a set of m graphs , the graph with is called the union of , denoted by .
A time signal for some positive integer m is called a piecewise constant switching signal with dwell time for some if there exists a time sequence satisfying ; for any positive integer k, ; , , for all . Given a node set and a piecewise constant switching signal , define a switching graph where for all . For a switching graph, let denote the neighbor set of node i at time instant t. Associated with a switching graph , the matrix is called a time-varying weighted adjacency matrix of if ; ; and otherwise. Let be such that and if . Then, is called the Laplacian of associated with .
3. Problem Statement
In this paper, we consider the distributed polynomial path tracking control problem for an AUV swarm system consisting of
N AUVs. As in [
38], we suppose the AUV is of full actuation. Then, for
, the dynamics of the
ith AUV take the following form
where
denote the generalized position, velocity and control input of the
ith AUV, respectively.
are unknown system parameters satisfying
with
and
denoting the nominal part and uncertain part of
, respectively,
. Without a loss of generality, suppose
. For
, let
denote the vector of uncertain parameters of the
ith AUV. Obviously,
if and only if
, i.e., there is no system uncertainty. In what follows, let
be a compact set containing the origin of
.
For
, define the state of the
ith AUV as
. Then, system (1) can be rewritten into the following compact form:
where
Here,
denotes an
n-dimensional identity matrix.
Moreover, define the nominal parts of the system matrices
and
as follows:
Since , it can be easily verified that is controllable.
Consider the following polynomial path
where
m is some non-negative integer, and
,
are constant vectors.
The distributed polynomial path tracking of the swarm system (2) is defined in the following way. For each AUV, denote the local formation command as
, which defines the relative position of the
ith AUV with respect to the polynomial path. Then, the absolute trajectory tracking error for the
ith AUV is defined by
Note that an important property of the polynomial path (
3) is that it can be generated by a virtual leader system in the following form:
where
with
Here, ⊗ denotes the Kronecker product of matrices, and denotes the internal state of the virtual leader. Note that is observable, and so is .
Remark 1. In this paper, we consider the case where the polynomial path cannot be known in advance by any AUV. Instead, it will be estimated in a distributed way by each AUV depending solely on neighboring information exchange by the distributed observer.
The communication network for the virtual leader system (5) and the swarm system (2) is described by a switching graph with and . Here, the node 0 is associated with the virtual leader, and the node i, is associated with the ith AUV. For , , if and only if the jth AUV can receive information from the ith AUV. Moreover, for , if and only if the ith AUV can receive information from the virtual leader. Let the weighted adjacency matrix of the digraph be . Define a subgraph of as with and . Let be the Laplacian of and .
The following assumptions are imposed on the communication graphs and , respectively.
Assumption 1. There exists a subsequence of satisfying for some , such that every node i, , is reachable from node 0 in the union graph .
Assumption 2. The switching graph is undirected. Moreover, there exists a subsequence of satisfying for some , such that the union graph contains a spanning tree.
Remark 2. Both Assumptions 1 and 2 are referred to as the jointly connected condition in the literature [32], where Assumption 1 is used for modeling the communication networks for leader–follower-type multi-agent systems, whereas Assumption 2 is used for modeling the communication networks for leaderless-type multi-agent systems. The jointly connected condition is possibly the mildest condition ever imposed on communication networks, and can tolerate the extreme case where the communication network topology is disconnected for all of the time instants. Now, we are ready to formulate the polynomial path tracking problem as follows.
Problem 1. Given systems (2), (5) and the communication graph , design a distributed control law in the following form:such that there exists for any , and for any system initial condition, there exists some constant vector , known or not, satisfying Remark 3. In this paper, it is assumed that the absolute positions s of the AUVs are not available. Therefore, it is, in general, impossible to drive to zero. While, in practice, the absolute position tracking is usually immaterial, it is the formation generation and keeping that matters. Thus, instead of regulating to zero, it suffices for the absolute tracking errors to converge to a common constant vector, which, in turn, implies the achievement of formation generation and keeping.
4. Main Results
The distributed robust control scheme proposed in this paper is composed of three parts. First, an output-based distributed observer was invoked to recover the polynomial path for each AUV. Second, a pseudo position estimator was adopted to generate the pseudo position for each AUV. Finally, based on the estimated polynomial path and the pseudo position, a certainty equivalent robust internal model control law was synthesized to achieve reference trajectory tracking. The details of these three parts are presented in sequence as follows.
First, we introduce the output-based distributed observer for the virtual leader.
Since
is observable, let
be the positive definite matrix solution to the following algebraic Riccati equation
Then, for
, design
where
and
are the estimates of
and
, respectively, and
is the observer gain.
For
, define
and
. Then,
For
,
, define the notation
. Let
. Then, we have
Using Theorem 4.1 of [
32], we have the following result.
Lemma 1. Given systems (5) and (12), under Assumptions 1 and 2, for any system initial condition, and any , both and tend to zero exponentially as .
Remark 4. The dynamic compensator (12) is called an output-based distributed observer of the virtual leader system (5) in the sense that: (1) it relies solely on the output of the virtual leader ; (2) it only requires local information of neighboring AUVs; (3) it can recover the information of the virtual leader for each AUV. In this way, the information of the virtual leader is transmitted to each AUV over the unreliable jointly connected switching network .
Next, we introduce the pseudo position estimator. For
, design
where
is the consensus gain.
Define
. Then, we have
Then, under Assumption 1, using Theorem 2.8 of [
1], it follows that, for any
,
exponentially for some static vector
.
Remark 5. For , defineand Then, using (17), it follows thatexponentially. Equation (18) means that, though the absolute position is not available, it is possible to generate a pseudo position for each AUV ensuring that the differences between the pseudo positions and the absolute positions of all of the AUVs are a common constant vector. It is this very result that leads to the solution to the polynomial path formation problem by a distributed control law in the form of (8). In terms of
, the system dynamics (1) become
Moreover, we define the new tracking error as
Then, if and only if . As a result, to solve Problem 1, it suffices to solve the following problem.
Problem 2. Given systems (5), (19) and the communication graph , design a distributed control law in the form of (8) such that there exists for any , and, for any system initial condition, To solve Problem 2, we will rewrite the system dynamics in a new compact form. First, define the following augmented exosystem:
Therefore,
. Then, letting
gives
where
Note that the minimal polynomials of
,
and
are the same. Therefore, let
and thus the following matrix pair
incorporates an
n-copy internal model of
.
Since all of the eigenvalues of
are zero, for any eigenvalue
of
, note that
gives
Thus, by Lemma 1.26 of [
39], the matrix pair
is able to be stabilized, where
Furthermore, let
be such that
is Hurwitz.
For
, let
and
. Therefore,
exponentially using (18). Now, we are ready to present the certainty equivalent robust internal model control law as follows:
The overall distributed robust control scheme proposed in this paper is composed of the distributed observer (12), the pseudo position estimator (15) and the certainty equivalent robust internal model control law (27). The information flow among different parts of the distributed robust control scheme is illustrated in
Figure 1. A pseudo code used to calculate the control input by the distributed robust control scheme for the
ith AUV,
, is given by Algorithm 1.
Algorithm 1 Calculating the control input by the distributed robust control scheme for the ith AUV |
Input:, , Output: |
1: According to the dimension of the position output n and the order of the polynomial path m, determine the matrix pairs (7) and (6). |
2: Solve the Riccati Equation (10) and design L using (11). |
3: Select any and implement the distributed observer (12). |
4: Select any and implement the pseudo position estimator (15) based on and . |
5: Determine using (24), using (25) and using (26). |
6: Select such that is Hurwitz. |
7: Based on from the distributed observer (12) and from the pseudo position estimator (15), implement the certainty equivalent robust internal model control law (27). |
The main result of this paper is presented as follows.
Theorem 1. Given systems (5), (23) and the communication graph , under Assumptions 1 and 2, Problem 2 is solvable by the control law composed of (12), (15) and (27) for any .
Proof. Define, for
,
Since
is Hurwits, there exists
, containing the origin of
such that
is Hurwitz for any
. Then, using Lemma 1.27 of [
39], the following matrix equation
have a unique solution pair
, which, in addition, satisfies
Substituting (27) into (23) gives
and
Next, for
, define
Then, using (28), we have
and
For
, define
. Then, it follows that
with
Since
is Hurwitz for any
, and
decay to zero exponentially as
, using Lemma 2.5 of [
32], it follows that
Moreover, using (29), it follows that
Therefore, using (36), and thus the proof is complete. □
5. Numerical Simulations
In this section, we will use numerical simulations to illustrate and validate the proposed control scheme. The control task is taken from [
38], where a fleet of AUVs executed the formation task in Monterey Bay during August 2003 to observe and predict ocean processes.
5.1. Aim of the Experiment
We begin with the aim of the experiment. In [
38], the aim of the experiment is to drive the AUV fleet to cruise in the ocean in a static triangular formation while following the reference trajectory of a straight line, illustrated by
Figure 2.
In this paper, we consider a similar task as in [
38]. Consider a swarm system of four AUVs, whose dynamics are given by
where
. For
, suppose the system parameters are given by
,
,
,
,
,
.
The polynomial path is given by
The local formation vector is given by
The communication network is shown by
Figure 3. In particular, the communication network
is assumed to switch among six subgraphs
, ⋯
, periodically every
sec. Suppose that
. It can be verified that Assumptions 1 and 2 are both satisfied. The distinguished characteristic of the communication network
is that it is disconnected the entire time.
5.2. Methodology
Next, we will follow Algorithm 1 to conceive the distributed robust control scheme.
- 1.
The dimension of the position output is 3, and the order of the polynomial path is 1. Therefore, we have
and
- 2.
The solution to the following Riccati equation
is
and thus
- 3.
Select
and design the distributed observer as follows
where
.
- 4.
Select
and design the pseudo position estimator as follows
where
.
- 5.
Using (24)
and, using (25),
Using (38), it follows that
and, thus, using (26),
- 6.
By letting
with
and
it follows that the eigenvalues of
are located at
that is,
is Hurwitz.
- 7.
Let
and the certainty equivalent robust internal model control law be designed as follows:
5.3. Results
Now, we examine the system performance using simulation results. Suppose that the initial positions and velocities of the AUVs are given by
The initial values of the control laws, i.e., the components of , and , take random values from the interval .
5.3.1. Standard Case
The simulation results are shown in
Figure 4,
Figure 5,
Figure 6 and
Figure 7. In particular, the performance of the distributed observer over the unreliable switching communication network
is shown in
Figure 4. It can be seen that the polynomial trajectory has been successfully recovered by the distributed observer of each AUV. The performance of the pseudo position estimator is shown in
Figure 5. As proved, differences between the actual positions and the pseudo positions of all of the AUVs will converge to a common constant vector, which, in our case, is very close to zero. The absolute tracking errors for all of the AUVs are shown in
Figure 6. It can be seen that these tracking errors also converge to a common constant vector, as required by the control objective (9), i.e., the distributed tracking problem has been successfully achieved by the proposed distributed robust control scheme. Finally, the 3D trajectories of all of the AUVs are plotted in
Figure 7, where the process of formation generation and keeping can be seen straightforwardly.
5.3.2. Comparative Study
In this case, we compare the proposed distributed robust control scheme with a typical existing work [
29] from the perspective of a communication network exclusively. For simplicity, for the control scheme proposed in [
29], it is assumed that the absolute position of the AUV is available for control feedback, and the system parameters are fully known, while keeping in mind that, for the control scheme proposed in this paper, the absolute position of the AUV is not available, and the system parameters are unknown. Suppose that the polynomial path considered in this case is given by
First, we consider the ideal static and connected communication network for the result in [
29]. Suppose the communication network is the union of the six subgraphs of
Figure 3, which is shown in
Figure 8. Under the communication graph
, the simulation results using the control method in [
29] are shown in
Figure 9. It can be seen that the tracking errors have been driven to zero asymptotically. Under the communication graph
defined by
Figure 3 with different switching period
, the simulation results using the control method in [
29] are shown in
Figure 10 and
Figure 11. When the communication network becomes unreliable and switched, the tracking errors will no longer converge to zero, which concludes that the control method in [
29] is effective for a static and connected communication network, but cannot effectively deal with an unreliable jointly connected switching communication network. Similar to the design process as in the standard case, we can design the distributed robust control scheme proposed in this paper for the new
given by (51). The simulation results using the proposed control scheme of this paper are given in
Figure 12 and
Figure 13, which show that successful tracking has been achieved for both cases.
6. Methodology Discussion
In this section, we will further examine the effectiveness of the proposed control scheme from the perspective of measurement noise. There are two sources of measurement noises associated with the distributed robust control scheme proposed in this paper.
The first one is the velocity measurement noise imposed on
. Suppose that
where
denote the measured velocity, true velocity and velocity measurement noise, respectively. In what follows, we will show, using simulation results, the effect of
on the system performance. In the simulations, suppose that the entries of
take random values uniformly from
, with
being the magnitude of the velocity measurement noise. Note that the velocity measurement noise will affect the pseudo position estimator (15) and the robust internal model control law (27). Simulation results with
are shown in
Figure 14 and
Figure 15, from which, it can be observed that, due to the velocity measurement noise, the differences between the pseudo positions and the actual positions of the AUVs will approximately converge to some common vector, while this common vector is not constant but time-varying. As a result, the steady state trajectory tracking errors of all of the AUVs also approximately converge to some time-varying common vector.
The second one is the relative position measurement noise imposed on
. In this scenario, suppose the pseudo position estimator takes the following form
where
denotes the lumped relative position measurement noise for the
ith AUV. Again, we will show, using simulation results, the effect of
on the system performance. Similarly, suppose that the entries of
take random values uniformly from
, with
being the magnitude of the relative position measurement noise. Note that the relative position measurement noise will affect the pseudo position estimator (15), and thus the tracking performance of the system. Simulation results with
are shown in
Figure 16 and
Figure 17. Similar to the case of velocity measurement noise, the differences between the pseudo positions and the actual positions of the AUVs will approximately converge to some time-varying common vector, which, in turn, makes the approximate common path tracking error time-varying too. Note that the gain for the distributed observer
will amplify the magnitude of the relative position measurement noise. As a result,
should not be selected as overly large in the presence of relative position measurement noise.
7. Conclusions
In this paper, a distributed robust control scheme is proposed to solve the polynomial path tracking problem for a swarm of uncertain AUVs facing three application challenges. First, the communication network is unreliable, satisfying merely the jointly connected condition. Second, only the relative position measurement between neighboring AUVs over the communication network is available for control feedback. Third, the second-order model dynamics for the AUV contain uncertain system parameters. To address these issues, three control parts were designed constituting the distributed robust control scheme, namely, the distributed observer, the pseudo position estimator and the certainty equivalent robust internal model control law. Comprehensive simulation results have validated the effectiveness of the proposed control scheme, especially the robustness against the unreliable and switching communication network when comparing with other existing results. Moreover, a further discussion on the effect of measurement noises on the system performance was conducted, where it was shown using simulation results that the proposed control scheme shows a certain resiliency with respect to velocity and relative position measurement noises. In this paper, it is assumed that the AUVs are fully actuated, which results in a linear system dynamic model. In the future, it would be interesting to further consider the case of underactuated AUVs with complex nonlinear system dynamics.